VISION TRACKING PREDICTION

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1 VISION RACKING PREDICION Eri Cuevas,2, Daniel Zaldivar,2, and Raul Rojas Institut für Informati, Freie Universität Berlin, ausstr 9, D-495 Berlin, German el División de Electrónica Computación, Universidad de Guadalajara Av Revolucion No 500, CP 44430, Guadalajara, Jal, Mexico el {cuevas, zaldivar, rojas}@inffu-berlinde ABSRAC: he Kalman filter has been used successfull in different prediction applications or state determination of a sstem One of the more important fields in computer vision is the object tracing Different movement conditions and occlusions can hinder the vision tracing of an object In this report we present the use of the Kalman filter as a predictor in vision tracing We consider the capacit of the Kalman filter to allow small occlusions and also the use of the extended Kalman filter to model complex movements of objects Introduction he celebrated Kalman filter, rooted in the state-space formulation or linear dnamical sstems, provides a recursive solution to the linear optimal filtering problem It applies to stationar as well as nonstationar environments he solution is recursive in that each updated estimate of the state is computed from the previous estimate and the new input data, so onl the previous estimate requires storage In addition to eliminating the need for storing the entire past observed data, the Kalman filter is computationall more efficient than computing the estimate directl from the entire past observed data at each step of the filtering process his paper is organized as follows In section 2 an introductor treatment of Kalman filter and its application in vision tracing is explained In section 3 the optimum estimation is deduced In section 4 the Kalman filter is analsed In section 5 the extended Kalman filter is presented In section 6 the vision tracing using the Kalman filter and it s extended version are proposed and tested Finall in section the conclusions are presented 2 Sstem dnamic Consider a linear, discrete-time dnamical sstem described b the bloc diagram shown in Figure he concept of state is fundamental to this description he state vector or simpl state, denoted b, is defined as the minimal set of data that is sufficient to uniquel describe the unforced dnamical behavior of the sstem; the subscript denotes discrete time In other words, the state is the least amount of data on the past behavior of the sstem that is needed to predict its future behavior picall, the state x is unnown o estimate it, we use a set of observed data, denoted b the vector In mathematical terms, the bloc diagram of Figure embodies the following pair of equations: Process equation where F +, + +, x x F x + w is the transition matrix taing the state from time to time + he process noise w is assumed to be additive, white, and Gaussian, with zero mean and with covariance matrix defined b Q for n E ww n (2) 0 for n where the superscript denotes matrix transposition he dimension of the state space is denoted b M Hx + v (3) where is the observable at time and is the H measurement matrix he measurement noise v is assumed to be addilive, white, and Gaussian, with zero mean and with covariance matrix defined b x

2 R for n E vv n (4) 0 for n Moreover, the measurement noise v is uncorrelated with the process noise w he dimension of the measurement space is denoted b N he Kalman filtering problem, namel, the problem of jointl solving the process and measurement equations for the unnown state in an optimum manner [] ma now be formall stated as follows: Use the entire observed data, consisting of the vectors, 2,,, to find for each the minimum mean- square error estimate of the state x w he problem is called filtering if i>, and smoothing if i< i, prediction if Process equation Measurement equation x + z - x H where E is the expectation operator he dependence of the cost function J on time 4 Kalman filter Suppose that a measurement on a linear dnamical sstem, described b Eqs and (3), has been made at time he requirement is to use the information contained in the new measurement to update the estimate of the unnown state x Let denote a priori estimate of the state, which is alread available at time With a linear estimator as the objective, we ma express the a posteriori estimate as a linear combination of the a priori estimate and the new measurement, as shown b ˆ G x + G (8) where the multipling matrix factors G and G are to be determined he state-error vector is defined b x x (9) Appling the principle of orthogonalit to the situation at hand, we ma thus write F+, v E x i 0 for i, 2,, (0) Figure Signal-flow graph representation of a linear, distrete-time dnamical sstem 3 Optimum estimation Suppose we are given the observable x + v (5) where x is an unnown signal and v is an additive noise component Let x denote the a posteriori estimate of the signal x, given the observations In, 2,, general, the estimate x is different from the unnown ˆ signal x o derive this estimate in an optimum manner, is need it that the cost function is nonnegative and nondecreasing function of the estimation error x defined b x x (6) hese two requirements are satisfied b the meansquare error [2]defined b 2 [( ˆ ) ] (7) J E x x J E x 2 [( ) ] Using Eqs (3), (5), and (6) in (7), we get E[( x G x G H x G v ) ] 0 for i,2, ˆ i Since the process noise w and measurement noise are uncorrelated, it follows that v E [ v ] 0 Using this relation and adding the element G x G x, we ma rewrite Eq (8) as [( ) i ( ) i ] 0 i E IG H G x + G x ( 9) where I is the identit matrix From the principle of orthogonalit, we now note that E[( x x ) ] 0 i Accordingl, Eq () simplifies to ( ) [ E i ] 0 for i, 2,, I G H G x (2 0) For arbitrar values of the state x and observable, i Eq (2) can onl be satisfied if the scaling factors G and are related as follows: G

3 IG H G 0 or, equivalentl, G is defined in terms of G as G IG H (3) Substituting Eq (3) into (5), we ma express the a posteriori estimate of the state at time as x + G ( H x ˆ ) in light of which, the matrix G^ is called the Kalman gain here now remains the problem of deriving an explicit formula for G Since, from the principle of orthogonalit, we have it follows that E[( x ) ] 0 (4) i E[( x ) ˆ ] 0 i where ˆ is an estimate of given the previous measurement Define the innovations, 2,, process ˆ (5) he innovation process represents a measure of the "new" information contained in ; it ma also be expressed as v + H x (6) Hence, subtracting Eq (4) from (3) and then using the definition of Eq (5), we ma write E[( x ) ] 0 (7) Using Eqs (3), we ma express the state-error vector x ˆ x as: x ( IG H ) x G v (8) Hence, substimting Eqs (6) and (8) into (7), we get E[{( IG H ) x G v }( H x + v )] 0 (9) Since the measurement noise v is independent of the state x and therefore the error, the expectation of x Eq (9) reduces to ( IG H ) E[ x x ] H G E[ v v ] 0 (20) Define the a priori covariance matrix P E[( x )( x ) ] E[ xx ] (2) hen, invoing the covariance definitions of Eqs (4) and (2), we ma rewrite Eq (20) as ( I G H ) P H G R 0 Solving this equation for G, we get the desired formula G P H [ H P H + R ] (22) where the smbol [] denotes the inverse of the matrix inside the square bracets Equation (22) is the desired formula for computing the Kalman gain, which is defined in terms of the a priori covariance matrix P o complete the recursive estimation procedure, we consider the error covariance propagation, which describes the effects of time on the covariance matrices of estimation errors his propagation involves two stages of computation: he a priori covariance matrix P at time is defined b Eq (2) Given P, compute the a posteriori covariance matrix P, which, at time, is defined b P E[ x x ] 2 Given the "old" a posteriori covariance matrix, P, compute the "updated" a priori covariance matrix P o proceed with stage, we substitute Eq (8) into (23) and note that the noise process is independent of the a priori estimation error x We thus obtain v G P ( IG H ) P ( I G H ) + G R G (24) Expanding terms in Eq (24) and then using Eq (22), we ma reformulate the dependence of the a posteriori covariance matrix on the a priori covariance matrix P P in the simplified form P ( IGH) P (25) For the second stage of error covariance propagation, we first recognize that the a priori estimate of the state is defined in terms of the "old" a posteriori estimate as follows: x F (26), We ma therefore use Eqs and (26) to express the a priori estimation error in et another form:

4 x x, F x + w (27) x F x + w + +, w Q Accordingl, using Eq (27) in (2) and noting that the process noise is independent of x we get w ˆ H x + v v R P F E[ x x ] F + E[ w w ],, F P F + Q (28),, which defines the dependence of the a priori covariance matrix P on the "old" a posteriori covariance matrix P With Eqs (26), (28), (22), (2), and (25) at hand, we ma now summarize the recursive estimation of state as shown in figure 2 his figure also includes the initialization In the absence of an observed data at time 0, we ma choose the initial estimate of the state as: x0 E[ x 0] and the initial value of the a posteriori covariance matrix as: P E[( x E[ x ])(( x E[ x ]) ] (30) his choice for the initial conditions not onl is intuitivel satisfing but also has the adventage of ielding an unbiased estimate of the state x Initialization State estimate State with measurement correction E[ x ] 0 0 P E[( x E[ x ])( x E[ x ] ) P F, P F, + Q G F, PH HPH + R + G ( H P ( IG H ) P 5 Extended alman filter he Kalman filtering estimate a state vector in a linear model of a dnamical sstem If, however, the model is nonlinear, we ma extend the use of Kalman filtering through a linearization procedure he resulting filter is referred to as the extended Kalman filter (EKF) [3-5] o set the stage for a development of the extended Kalman filter, consider a nonlinear dnamical sstem described b the state-space model x f + (, x ) + w (3) h(, x ) + v (32) where, w and are independent zero-mean white v Gaussian noise processes with covariance matrices R and respectivel Here, however, the functional Q f(, x ) denotes a nonlinear transition matrix function that is possibl time-variant Liewise, the functional h(, x ) denotes a nonlinear measurement matrix that ma be time-variant, too Figure 2 Sumar of the Kalman filter Once a linear model is obtained, the standard Kalman filter equations are applied he approximation proceeds in two stages Stage he following two matrices are constructed: F +, H f(, x ) x h(, x ) x hat is, the ijth entr of F +, x x (33) (34) is equal to the partial derivative of the ith component of F (, x) with respect to the th component of x Liewise, the ijth entr of H is equal to the partial derivative of the ith component of H(,) x with respect to the jth component of x In the former case, the derivatives are evaluated at x ˆ, while in the latter case, the derivatives are evaluated at x ˆ he

5 entries of the matrices F and H are all nown (ie, +, computable), b having and x available at time Stage 2 Once the matrices F and H are evaluated +, the are then emploed in a first-order alor approximation of the nonlinear functions F(, x) and H(, x) around x and x, respectivel ˆ ˆ With the above approximate expressions at hand, we ma now proceed to approximate the nonlinear state equations (3) and (32) as shown b, respectivel, ˆ x+ F+, x + w + d Hx + v where we have introduced two new quantities: { (, ˆ ) ˆ hxx Hx } (35 58) d f( x, ) F (36 59) +, he entries in the term are all nown at time, and, therefore, can be regarded as an observation vector at time n Liewise, the entries in the term d are all nown at time Given the linearized slate-space model of Eqs (35) and (36), we ma then proceed and appl the Kalman filter theor of section 4 to derive the extended Kalman filter 6 Kalman vision tracing sstem he main application of the Kalman filter in robot vision is the object tracing As input is used a sequence of images captured b a camera hen the object is segmented and later it s positions is calculated herefore, we will tae as sstem state x the x and position of the object in the instant hus, we can use the Kalman filter in a defined search window centered in the predicted filter value he steps to use the Kalman filter for vision tracing are: previous stage x ˆ ) and we use its real position (measurement) to estimated the state correction using the Kalman filter, finding x ˆ o tests the performance of the Kalman filter in vision tracing, the tracing of a soccer ball is chosen and the following cases are considered: a) he ball realizing a lineal uniform movement is traced, which could be described b the following sstem equations: x F x + w + +, x+ 0 0 x w x x Hx + v x xm m v x In the figure 3 the prediction of the object position is shown for each instant as well as the real trajector b) Occlusion test If during the object localization the object is not in the neighborhood of the predicted state, the object is considered hidden b some other object, then the measurement correction is not calculated and the filter prediction is tae as position he figure 4 shows the filter performance during the object occlusion he sstem was modeled with the same equations used in the previous case Initialization (0) In this stage is necessar looed for the object in the whole image, due we do not now the previous object position We obtain this wa x 0 Also we can considerer initiall a big error tolerance ( P ) 0 2 Prediction (>0) In this stage, we predict the relative position of the object he position is considered as search center to find the object 3 Correction (>0) In this part we locate the object (which is in the neighborhood point predicted in the Figure 3 Position prediction using the Kalman filter

6 x xm m v x function to drift bodil he effect of an external observation is to superimpose a reactive effect on the diffusion in which the densit tends to pea in the vicinit of observations Conclusions Figure 4 Kalman filter performance during the occlusion c) Figure 5 shows the performance of the EKF to trac a complex trajector versus the performance of a Kalman filter he Kalman filter in vision tracing, provides a recursive solution to the linear optimal prediction problem It is applicable to stationar as well as nonstationar environments he solution is recursive and each updated estimation of the state is computed from the previous estimation and the new input data requires onl the previous estimation he alman filter is restricted to tracing of objects whose movement can be modeled b a lineal sstem (not-complex movements) however the extended Kalman filter can be used as prediction algorithm in cases where the model is nonlinear Although the linealization of the equations supposes an approximation to the prediction problem, that, in vision tracing applications can be considered enough Due to the wor phases of the Kalman filter, prediction and correction, it is possible to use it to tolerate small occlusions presented in the objects, b considering in occlusion moments onl the predicted filter value References Figure 5 racing of a complex movement using an EKF d) For the extended Kalman filter the dnamic sstem was modeled using the unconstrained Brownian motion equations: x f + (, x ) + w ( + ) ( + ) exp( / 4 x) exp( / 4 ) x+ exp / 4( x 5 x) + exp / 4( 5 ) +w x+ + h(, x) + v while for the Kalman filter, the sstem was modeled using the equations of case a) [] RE Kalman, "A new approach to linear filtering and prediction problems", ransactions of the ASME, Ser D, Journal of Basic Engineering, 82, (960) [2] HL Van rees, Detection, Estimation, and Modulation heor, Part I New Yor, Wile 968 [3] AH Jazwinsi, Stochastic Processes and Filtering heor New Yor, Academic Press, 970 [4] PS Mabec, Stochastic Models Estimation and Control, Vol New Yor, Academic Press, 979 [5] PS Mabec, Stochastic Models Estimation and Control, Vol 2 New Yor, Academic Press, 982 [6] Nischwitz A, Haberäcer P, Masterurs Computergrafi und Bildverarbeitung, Berlin, Vieweg 2004

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